Path Integral Perturbation Theory

The importance of the path integral formalism should be clear by now and we have seen how it can be useful to obtain matrix elements and expectation values. However, its usefulness may not be very obvious when it is used for interacting particles. For such an interaction, when the non-Gaussian part of the action is weak compared to the quadratic part, then one expects perturbation expansion to be applicable. This is really much easier than in the operator formalism since we deal only with c-numbers in the path integration.

Often in quantum mechanics we may be confronted with systems that are very difficult to solve exactly. This implies that for such systems the Hamiltonian may not be diagonalized exactly and so we resolve to approximate solutions that may sometimes give results close to true experimental values. In quantum mechanics as in classical mechanics there are relatively few systems of physical interest for which the equations of motion may be solved exactly. Approximate methods are thus expected to play an important role in virtually all applications of the theory. For quantum mechanics, an exact solution of the Schrodinger equation exists only for few idealized problems. The perturbation theory is applied to those cases in which the real system can be described by a small change in an easily solvable idealized system.

In the evaluation of path integration, when the exponential functions are quadratic in q and q then the integral is exactly solvable. But the alternative method, the so-called perturbation theory, can be employed as mentioned. For that reason, we rewrite the transition amplitude 4.82:

where the Lagrangian L(q,q,t):

From the Lagrangian 11.2, we show that the transition amplitude 4.82, in which the potential U(q) is inherent, can be viewed as a sequence of interactions with the potential separated by free propagations.

It is useful further to represent the perturbation series pictorially by denoting the full transition amplitude with a double undirected straight line:

the unperturbed transition amplitude by undirected straight line: and insertion of the perturbation by a wiggly line:

ЛЛЛЛЛЛЛЛЛ.

Let us select the potential V₀(q) then we write the following transition amplitude for that potential: and diagrammatically represented:

To find the transition amplitude for subsequent potentials then we introduce the so-called perturbation v(q)

then follows the transition amplitude: or

that we represent pictorially by a double line:

We expand the second exponential in series of the perturbation v(q) (Neumann-Liouville expansion or Dyson series):

or

that we represent pictorially (Figure 11.1):

So, the full amplitude can be written as sum of partial amplitudes in which the particle is not scattered plus scattered once plus scattered twice and so on. Here, for the zero order of the perturbation theory

FIGURE 11.1 Pictorial representation of the full transition amplitude as sum of partial amplitudes in which the particle is not scattered plus scattered once plus scattered twice and so on.

is the contribution to the transition amplitude resulting from the particle going from q_a to q_b in the time (t_b - t_a), without being affected by the perturbation v(q). For the first order of the perturbation theory then

which is the contribution to the transition amplitude when the particle is scattered once due to the perturbation v(q) (Figure 11.2).

The last factor of 11.10 is in the discretized form:

where i indicates the sum over time slices and then the transition amplitude where the normalization factor is

FIGURE 11.2 Illustrating the time-ordering procedure in 11.7 or 11.8 and showing the initial, final and intermediate time moments together with the respective coordinates during the evolution of the system.

Consider that the particles travel from point q_a at the time moment t_a via point q_s with time moment s to point q_b with time moment t_b:

From here and 11.12, the first order perturbation may be written as an integral equation for the transition amplitude:

or

where

The pictorial representation is as follows (Figure 11.3)

For the second order of the perturbation theory we consider first the following Figure 11.4 (shaded area). From Figure 11.4, we have

FIGURE 11.3 Pictorial representation of the first order perturbation theory.

FIGURE 11.4 Representation of the second order of the perturbation theory that can be evaluated by the shaded area.

From these time intervals then or or

From here and Figure 11.4 we observe that the integrals covering the triangle above the diagonal of the square and the triangle below the diagonal of the square are equal:

We can therefore write the second order of the perturbation theory as follows:

The pictorial representation is as in Figure 11.5.

This procedure may be continued for higher orders of the perturbation theory and we have

FIGURE 11.5 Pictorial representation of the second order of the perturbation theory.

We rewrite this as follows:

This may be rewritten in the following form by iteration:

So, for K_v we have the integral form

This mimics the usual equation for the time-evolution operator in the interaction picture if one writes equation 11.27 in operator form. The pictorial representation that is the full transition amplitude equals unscattered transition amplitude plus sum of all processes with the last scattering at time t_c (Figure 11.6):

This integral equation is linked with the transition amplitude via a given potential. If v is the quantity then the perturbation series is limited to only a few terms, and we can use 11.27 to obtain a similar equation for wave function:

or

The first summand is an unchanging interaction (incident wave) and the second a scattering wave. Due to the Bloch scattering theory we swap K,, for K₀ and

Considering

FIGURE 11.6 Pictorial representation of the full transition amplitude equals unscattered transition amplitude plus sum of all processes with the last scattering at time t_r

and the fact that states have delta-normalization as seen earlier, then

and so

then from we have

Here, 4/₀(qi„tb) *^s the unperturbed wave function satisfying the unperturbed Schrodinger equation and the transition amplitude satisfying the relation Here, в is the Heaviside step function which permits equation 11.35 to be rewritten as follows:

From here, considering 11.37 then

or

or

So, equation 11.39 is an integral equation for i//(q_b,t_b) and imitates the Schrodinger equation.